Thank you for all these details.
I don't get what could be wrong, there is no sensitive code in the files. It does not use any of the functions you disabled (unless there is some dependency I did not see).
I would suggest you re-enable the functions you disabled, see if the component works. If it does, disable each function one by one until the component does not work anymore. That will narrow the search and it will be easier to look for where the trouble comes from.
If the component does not work with all your php functions enabled again, we have to look somewhere else. Note that I have the free version of admin tools on my test site and the component works as expected.
One more thing: the stylemaster file also loads other style files that are found in its sub-folders. They all need to have same permissions. Just double-checking.

Thanks for the file. Your restrictions are extremely tight and it should not be a problem to pass parameters to a file through regular requests.

I am going to see how I can create the stylesheets without having this kind of CSS/PHP file. It seemed like a good idea at first, but I want to avoid these permissions issues as much as possible. There is no reason why you should not be able to secure your site the way you did. I have some idea on how to do that. Would you be willing to test it once I came up with it? Thank you.